{
  "id": "2401.08071",
  "version": "v1",
  "published": "2024-01-16T02:54:44.000Z",
  "updated": "2024-01-16T02:54:44.000Z",
  "title": "On free boundary problems shaped by oscillatory singularities",
  "authors": [
    "Damião Araújo",
    "Aelson Sobral",
    "Eduardo V. Teixeira",
    "José Miguel Urbano"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We start the investigation of free boundary variational models featuring oscillatory singularities. The theory varies widely depending upon the nature of the singular power $\\gamma(x)$ and how it oscillates. Under a mild continuity assumption on $\\gamma(x)$, we prove the optimal regularity of minimizers. Such estimates vary point-by-point, leading to a continuum of free boundary geometries. We also conduct an extensive analysis of the free boundary shaped by the singularities. Utilizing a new monotonicity formula, we show that if the singular power $\\gamma(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,\\delta}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-16T02:54:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "35J75"
    ],
    "keywords": [
      "free boundary problems",
      "variational models featuring oscillatory singularities",
      "free boundary variational models",
      "boundary variational models featuring oscillatory",
      "singular power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}